Integration. Change of Variables.

[sepoto]

\(\displaystyle \frac{1}{\sqrt{2\pi}}\int^x_0 e^{-u^2/2}du=\frac{1}{\sqrt{2\pi}}\int^{x/\sqrt{2}}_0 e^-{t}^{2}*\sqrt{2}dt\)

Right now I am questioning how the x became an \(\displaystyle x/\sqrt{2}\) on the integral on the right. Also for the integral on the right there is out front a \(\displaystyle \frac{1}{\sqrt{2\pi}}\) where there wasn't before. So It looks like du on the left was replaced with \(\displaystyle \sqrt{2}\) on the right. Is that correct? I'm having some trouble understanding the equivalency of the two integrals.

Thanks for any responses...

 

[sepoto]

\(\displaystyle \int^{x/\sqrt{2}}_0 e^{-t}^{2}\)

The \(\displaystyle x/\sqrt{2}\) appears to me to be the x from the previous integral divided by du. I hope this is correct. Also the very last step of the problem where the limit is evaluated I don't understand. The manual says:

\(\displaystyle \lim_{x \to +\infty}=\frac{1}{\pi}*\frac{\sqrt{\pi}}{2}=\frac{1}{2}\)

I don't understand how a limit like that was derived.